Recent advances in neural networks have proven immensely successful for regression analysis, image classification, time series modeling, and so on [1]}. Neural Networks are models of the human brain and vision [2]}, [3]}. A neural network performs regression analysis, image classification, and time series modeling by performing a series of sequential operations, known as layers. Each of these layers is composed of neurons that are connected to neurons of other (typically, adjacent) layers. We consider a neural network with \(L+1\)  layers, where the input layer is layer 0, the output layer is layer \(L\) , and the number of nodes in layer \(l~(l=0,1,\ldots ,L)\)  is \(n_l\in \mathbb {N}\) . Let \(f^{(l)}:\mathbb {R}^{n_l}\rightarrow \mathbb {R}^{n_{l+1}}\)  be the function of each layer. The output of each layer is, therefore, a vector in \(\mathbb {R}^{n_{l+1}}\) . If the input data is \(\xi \in \mathbb {R}^{n_0}\) , then, at each layer, we have
\(\left\lbrace \begin{aligned}x^{(l+1)} &= f^{(l)}(x^{(l)}), & l=0,1,\ldots ,L-1, \\x^{(0)} &= \xi . &\end{aligned}\right.\) 
